MATH 160 Study Guide for Chapter 5 Integration
Fall, 2008
This Study Guide describes everything from Chapter 5 that you will be expected to know, understand, and be able to
do on Exam 3 and/or the final exam. Exam questions from Chapter 5 will ask you to do one or more of the tasks
described in this Study Guide. Of course, to do the things listed here you must be able to do almost everything listed
in previous Study Guides. Don’t be surprised if some exam questions require you to use ideas or skills from earlier
in the course.
Sec 5.1 Estimating with Finite Sums
__ 1. Use finite sums to calculate reasonable estimates for
• the area of a region enclosed by the graph of a positive-valued function y =f(x), the x-axis, and vertical
lines y = a and y = b, and
• the distance a body travels during a given time interval from a graph of its velocity.
Representative homework problems: Select from pages 333 – 335; #1 – 8, 12. Page 388; #1, 2.
__ 2. • From a table giving the velocity of a moving body at various times (not necessarily equal spaced), calculate
a reasonable estimate for the displacement (change in position) of the body over a given time interval.
(You are expected to use a calculator.)
• Given an equation v = f( t) for a velocity function (whose antiderivative can’t be found easily), generate a
table that gives the velocity of the moving body at various (equally spaced) times, and use this table to
calculate a reasonable estimate for the displacement of the body over a given time interval.
(You are expected to use a calculator.)
Explain why one might expect your calculations to produce a reasonable estimate for the displacement.
Representative homework problems: Select from pages 333 –334; #9 – 14.
Sec 5.2 Sigma Notation and Limits of Finite Sums
n
__ 1. A sum of the form SP =
 f (ck )xk
is called a Riemann sum for f on the interval [a, b].
k 1
Use words and pictures to explain how the interval [a, b] enters into the Riemann sum and what each of the
n
symbols n,

, P, f , xk, xk, and ck in this expression means.
k 1
__ 2. Given a function y = f( x) defined on an interval [a, b] and given a partition a = x0 < x1 < … < xn-1 < xn = b
of the interval, choose appropriate evaluation points c1, c2, c3, … , cn , write an expression for the associated
Riemann sum (in expanded form), and calculate the numerical value of the Riemann sum.
(Proficiency with summation notation is not required, but expect to see summation notation in later courses.)
Representative homework problems: Select from pages 342 – 343; #29 – 32.
Study suggestion: Calculator Lab on Riemann Sums.
__ 3. Write an expression/equation that defines the Riemann integral in terms of Riemann sums.
Explain (using words and pictures) what this definition means and why it is necessary to take the limit as n
approaches infinity in this definition.
Explain why the variable of integration in
b
a f (x) dx
is called a dummy variable.
__ 4. Use a Riemann sum to calculate a reasonably accurate approximate value for a given (definite) integral.
(You are expected to use a calculator.)
Study suggestion: Calculator Lab on Riemann Sums.
Sec 5.3 The Definite Integral
__ 0. Use standard area formulas for triangles, rectangles, quarter circles, half circles, etc. to evaluate definite
integrals by identifying the integral with the area of a familiar shape. (You must to know these standard
formulas.)
__ 1. Use properties of the definite integral summarized in Table 5.3 entries #3 - #5 to evaluate definite integrals
by rewriting the integral as a combination of integrals that either have given values or can be evaluated by
interpreting the integral as (positive or negative) area.
Representative homework problems: Pg 353; select from odd numbered problems from #9 – 26.
__ 2. Estimate integrals by using inequalities #6 and #7 in Table 3.5.
Suggested problems: Select from page 355; #63 – 72. (In #63 & 64 assume a < b. In #68 the integral
should be from 0 to 1, not 1 to 0.)
October 31, 2008
Sec 5.4 The Fundamental Theorem of Calculus
__ 1. Use a calculator to evaluate a given function F(x) =
x
a f (t) dt
for various values of x.
Use the resulting values to sketch a graph of the function y = F(x).
Study suggestion: Calculator Lab on the Fundamental Theorem of Calculus.
__ 2. Use a calculator to compute an approximate value for the derivative F ′ (c) of a given function
F(x) =
x
a f (t) dt
at a specified point x = c from the definition of the derivative of a function at a point.
Interpret the calculation in terms of the graph of the integrand y = f( x).
Study suggestion: Calculator Lab on the Fundamental Theorem of Calculus.
__ 3. Write a differentiation formula for functions F( x) =
x
a f (t) dt
defined by an integral with continuous
integrand. Explain why it is appropriate to call this theorem The Fundamental Theorem of Calculus. (This
theorem is on page 358 of the textbook where it is called The Fundamental Theorem of Calculus Part 1).
Use this formula to compute derivatives of functions defined by integrals with the independent variable as
the upper limit of integration.
Representative homework problems: Page 365; #31 & 32. Page 390; #77.
__ 4. Use the differentiation formula from #3 with the Chain Rule and properties of the integral to calculate the
derivatives of functions F( x) =
g ( x)
a
f (t) dt and F( x ) =

b
f (t ) dt
where the upper or lower limit of
g ( x)
integration is a differentiable function y = g(x).
Representative homework problems: Page 365; #27 – 36. Page 367; #63 & 64. Page 390; #78, 79, 80
__ 5. Use the differentiation formula from #3 (in reverse) to write an expression for an antiderivative of a given
continuous function and to solve initial value problems (and to show your answers are correct).
Suggested problems: Select from page 366; # 47 – 54
__ 6. Explain the difference between an indefinite integral
 f (x) dx
and a definite integral
b
a f (x) dx .
State a Theorem that gives a connection between indefinite integrals (antiderivatives) and definite integrals.
Show how this result follows from The Fundamental Theorem of Calculus Part 1 (from #3 above).
(In the textbook this result is called The Fundamental Theorem of Calculus Part 2.)
__ 7. Infer information about the motion of an object moving a long a coordinate axis from a graph of its velocity.
Representative homework problems: Page 366; #59 & 60. Page 388; #1, 2.
__ 8. Use The Fundamental Theorem of Calculus Part 2 to evaluate definite integrals.
Suggested problems: Page 365; #1 – 26.
EN D O F M AT E RI A L C O V ER E D O N E X AM 3
Exam 3 covers all of Chapter 4 as described in the Chapter 4 Study Guide and Sections 5.1– 5.4 of Chapter 5.
October 31, 2008
Sec 5.5 Indefinite Integrals and Substitution
__ 1. Write an integration formula corresponding to the Chain Rule.
Explain how integration by substitution is used to apply (or implement) this integration formula.
Study suggestion: Pages 368 – 371 of the text.
Representative homework problems: Pages 374 – 375; #1 – 12. Page 376; #61, 62.
__ 2. Evaluate indefinite integrals by u-substitution.
After you have evaluated an indefinite integral by u-substitution, identify the functions f and g that
connect your work with the Chain Rule. Express the u you used in the substitution process in terms of one
or more of the functions f and g and their derivatives.
Representative homework problems: Select from pages 374 – 375; #13 – 52.
__ 3. Solve initial value problems that require u-substitution method to find the required antiderivatives.
Representative homework problems: Pages 375 – 376; #53 – 60.
NOTE: You will not be asked to derive or to remember the integration formulas for sin 2θ and cos2θ (pg 373) on
the Final Exam. These two integration formulas will be given if needed to answer an exam question.
Sec 5.6 Substitution and Area between Curves
__ 1. Use change of variables (u-substitution), including changes in the limits of integration, to evaluate definite
integrals.
Representative homework problems: Pages 383; #1 – 24.
__ 1. Rewrite definite integrals involving arbitrary functions (indicated only by function notation), including
changes in the limits of integration, by making a change of variables (that is, a u-substitution). (You might
or might not be able to evaluate the resulting integral.)
Representative homework problems: Pages 386 – 387; #83, 84, 87, 88, 89.
__ 3. Use a definite integrals to calculate the area of a region enclosed by the graph of a given function y =f(x),
the x-axis, and vertical lines y = a and y = b. (Compare item #1 under Sec. 5.3 above.)
Representative homework problems: Page 365 – 366; #37 – 46 and #55. Page 384 – 385; #25 – 29.
__ 4. Given a description of a region bounded by the graphs of continuous functions and (perhaps) vertical and/or
horizontal lines, find the area of the region by
• sketching the region and deciding whether it is better to integrate with respect to x or with respect to y,
• sketching a representative area element used to find the integral for the area,
• writing the definite integral (or integrals) that represents the area, and
• evaluating the integral(s).
Representative homework problems: Pages 384 – 386; #30 – 80.
October 31, 2008